Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor,II

In our previous report [Aita, T., Morinaga, S., Hosimi, Y., 2004. Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor I. Bull. Math. Biol. 66, 1371–1403], an analogy between thermodynamics and adaptive walks on a Mt. Fuji-type fitness landscape in an artificial selection system was presented. Introducing the ‘free fitness’ as the sum of a fitness term and an entropy term and ‘evolutionary force’ as the gradient of free fitness on a fitness coordinate, we demonstrated that the adaptive walk (=evolution) is driven by the evolutionary force in the direction in which free fitness increases. In this report, we examine the effect of various modifications of the original model on the properties of the adaptive walk. The modifications were as follows: first, mutation distance d was distributed obeying binomial distribution; second, the selection process obeyed the natural selection protocol; third, ruggedness was introduced to the landscape according to the NK model; fourth, a noise was included in the fitness measurement. The effect of each modification was described in the same theoretical framework as the original model by introducing ‘effective’ quantities such as the effective mutation distance or the effective screening size.     

Thermodynamical interpretation of an adaptive walk on a Mt. Fuji-type fitness landscape: Einstein relation-like formula holds in a stochastic evolution

We have theoretically studied the statistical properties of adaptive walks (or hill-climbing) on a Mt. Fuji-type fitness landscape in the multi-dimensional sequence space through mathematical analysis and computer simulation. The adaptive walk is characterized by the "mutation distance" d as the step-width of the walker and the "population size" N as the number of randomly generated d-fold point mutants to be screened. In addition to the fitness W, we introduced the following quantities analogous to thermodynamical concepts: "free fitness" G(W) is identical with W+T x S(W), where T is the "evolutionary temperature" T infinity square root of d/lnN and S(W) is the entropy as a function of W, and the "evolutionary force" X is identical with d(G(W)/T)/dW, that is caused by the mutation and selection pressure. It is known that a single adaptive walker rapidly climbs on the fitness landscape up to the stationary state where a "mutation-selection-random drift balance" is kept. In our interpretation, the walker tends to the maximal free fitness state, driven by the evolutionary force X. Our major findings are as follows: First, near the stationary point W*, the "climbing rate" J as the expected fitness change per generation is described by J approximately L x X with L approximately V/2, where V is the variance of fitness distribution on a local landscape. This simple relationship is analogous to the well-known Einstein relation in Brownian motion. Second, the "biological information gain" (DeltaG/T) through adaptive walk can be described by combining the Shannon's information gain (DeltaS) and the "fitness information gain" (DeltaW/T).     

Quantifying slow evolutionary dynamics in RNA fitness landscapes

We re-examine the evolutionary dynamics of RNA secondary structures under directional selection towards an optimum RNA structure. We find that the punctuated equilibria lead to a very slow approach to the optimum, following on average an inverse power of the evolutionary time. In addition, our study of the trajectories shows that the out-of-equilibrium effects due to the evolutionary process are very weak. In particular, the distribution of genotypes is close to that arising during equilibrium stabilizing selection. As a consequence, the evolutionary dynamics leave almost no measurable out-of-equilibrium trace, only the transition genotypes (close to the border between different periods of stasis) have atypical mutational properties.     

Analytical solution of the evolution dynamics on a multiplicative-fitness landscape

In an infinite population the frequency distribution of individuals carrying a given number of mutations obeys a set of recursion equations, the equilibrium solution of which describes the mutation-selection balance. Although this solution is well-known in the case of a multiplicative-fitness landscape, where it is assumed that all mutations are deleterious and that each new mutation reduces the fitness of the individual by the same fraction, we are not aware of the existence of an analytical solution for the full dynamics. Using the generating function approach, we present here an explicit analytical solution for the frequency distribution recursion equations valid for all generations and initial conditions.This research was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Proj. No. 99/09644-9. The work of J.F.F. was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).     

Evolution in a flat fitness landscape

A simple model of a population of asexually reproducing individuals, evolving in a flat fitness landscape, is defined. It is shown that the model is equivalent to a dynamical system with stochastic dynamics, the Annealed Random Map Model. Thus, it is possible to solve exactly for the genealogy statistics and for the genetic variability of the population. Fluctuations of quantities, like the average relatedness and the variability, which also take place in the limit of an infinitely large population, are computed.     

Genealogical process on a correlated fitness landscape

We study with extensive numerical simulation the genealogical process of 2N haploid genetic sequences. The sequences are under selective pressure, and fitness values are assigned at random, but with a tunable degree of correlation to the fitness values of closely related sequences. The genealogies that we observe can be classified into three different categories, corresponding to different regimes of the mutation rate. At low mutation rates, the sequences remain localized around a small number of central sequences, which leads to trees with short pairwise distances and slow turnover of the most recent common ancestor of the population. At high mutation rates, we observe trees similar (but not identical) to those of neutral evolution. In this regime, the population drifts rapidly, and selection does not influence the distribution of fitness values in the population. The third regime, for intermediate mutation rates, is only found in strongly correlated landscapes. It resembles the one for high mutation rates in that the population drifts rapidly, but nevertheless selection still shapes the distribution of fitness values.     

Some reflections on evolutionary theories,With a classification of fitness

Using a classical life history model (the Smith & Fretwell model of the evolution of offspring size), it is demonstrated that even in the presence of overwhelming empirical support, the testability of predictions derived from evolutionary models can give no guarantee that the underlying fitness concept is sound. Non-awareness of this problem may cause considerable justified but avoidable criticism. To help understanding the variable use of fitness in evolutionary models and recognizing potentially problematic areas which need careful consideration, a hierarchical classification of definitions of fitness used in evolutionary models is presented. As a conclusion, it is advocated to use the term fitness more conscientiously than currently often practised and to think more about ways to develop fitness-free evolutionary theories compatible with Darwin's ideas.     

Interaction rates,vital rates,background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics

In this paper we are concerned with how aggregated outcomes of individual behaviours, during interactions with other individuals (games) or with environmental factors, determine the vital rates constituting the growth rate of the population. This approach needs additional elements, namely the rates of event occurrence (interaction rates). Interaction rates describe the distribution of the interaction events in time, which seriously affects the population dynamics, as is shown in this paper. This leads to the model of a population of individuals playing different games, where focal game affected by the considered trait can be extracted from the general model, and the impact on the dynamics of other events (which is not neutral) can be described by an average background fertility and mortality. This leads to a distinction between two types of background fitness, strategically neutral elements of the focal games (correlated with the focal game events) and the aggregated outcomes of other interactions (independent of the focal game). The new approach is useful for clarification of the biological meaning of concepts such as weak selection. Results are illustrated by a Hawk–Dove example.     

Genomic landscape and evolutionary dynamics of mariner transposable elements within the Drosophila genus

Background

The mariner family of transposable elements is one of the most widespread in the Metazoa. It is subdivided into several subfamilies that do not mirror the phylogeny of these species, suggesting an ancient diversification. Previous hybridization and PCR studies allowed a partial survey of mariner diversity in the Metazoa. In this work, we used a comparative genomics approach to access the genus-wide diversity and evolution of mariner transposable elements in twenty Drosophila sequenced genomes.

Results

We identified 36 different mariner lineages belonging to six distinct subfamilies, including a subfamily not described previously. Wide variation in lineage abundance and copy number were observed among species and among mariner lineages, suggesting continuous turn-over. Most mariner lineages are inactive and contain a high proportion of damaged copies. We showed that, in addition to substitutions that rapidly inactivate copies, internal deletion is a major mechanism contributing to element decay and the generation of non-autonomous sublineages. Hence, 23% of copies correspond to several Miniature Inverted-repeat Transposable Elements (MITE) sublineages, the first ever described in Drosophila for mariner. In the most successful MITEs, internal deletion is often associated with internal rearrangement, which sheds light on the process of MITE origin. The estimation of the transposition rates over time revealed that all lineages followed a similar progression consisting of a rapid amplification burst followed by a rapid decrease in transposition. We detected some instances of multiple or ongoing transposition bursts. Different amplification times were observed for mariner lineages shared by different species, a finding best explained by either horizontal transmission or a reactivation process. Different lineages within one species have also amplified at different times, corresponding to successive invasions. Finally, we detected a preference for insertion into short TA-rich regions, which appears to be specific to some subfamilies.

Conclusions

This analysis is the first comprehensive survey of this family of transposable elements at a genus scale. It provides precise measures of the different evolutionary processes that were hypothesized previously for this family based on PCR data analysis. mariner lineages were observed at almost all “life cycle” stages: recent amplification, subsequent decay and potential (re)-invasion or invasion of genomes.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2164-15-727) contains supplementary material, which is available to authorized users.     

Adaptedness, evolution and a hierarchical concept of fitness.

Fitness is a consequence of the adaptedness of an entity to its environment. The fitness of an evolutionary entity, when the entity is defined by a set of attributes determining how it interacts with its environment, is manifested as persistence of those attributes. Two measures of fitness are presented to explicate this concept: (1) an extensive measure of the persistence of the original set of attributes in the individual entity and its descendants, but corrected for evolution; this includes as a special case fitness as used in population genetics; and, (2) an intensive measure that is independent of the abundance of descendant exemplars. Fitness as conceived here is a function of survival time, degree of evolution, and when applicable, reproductive contribution. The rate of fitness increase of an entity will vary inversely with the degree of evolutionary change experienced by the entity, its descendants, or both through time. Adaptive evolution can increase the length of time that fitness accumulates by increasing the survival of descendants possessing all or some of the ancestral attributes. Reproduction, where possible, can increase the number of descendants. This concept of fitness is simple, unambiguous, coherent, and applicable to entities at any hierarchical level of interest to ecologists and evolutionary biologists.     

Predictability of evolutionary trajectories in fitness landscapes

Experimental studies on enzyme evolution show that only a small fraction of all possible mutation trajectories are accessible to evolution. However, these experiments deal with individual enzymes and explore a tiny part of the fitness landscape. We report an exhaustive analysis of fitness landscapes constructed with an off-lattice model of protein folding where fitness is equated with robustness to misfolding. This model mimics the essential features of the interactions between amino acids, is consistent with the key paradigms of protein folding and reproduces the universal distribution of evolutionary rates among orthologous proteins. We introduce mean path divergence as a quantitative measure of the degree to which the starting and ending points determine the path of evolution in fitness landscapes. Global measures of landscape roughness are good predictors of path divergence in all studied landscapes: the mean path divergence is greater in smooth landscapes than in rough ones. The model-derived and experimental landscapes are significantly smoother than random landscapes and resemble additive landscapes perturbed with moderate amounts of noise; thus, these landscapes are substantially robust to mutation. The model landscapes show a deficit of suboptimal peaks even compared with noisy additive landscapes with similar overall roughness. We suggest that smoothness and the substantial deficit of peaks in the fitness landscapes of protein evolution are fundamental consequences of the physics of protein folding.     

Source-sink dynamics shape the evolution of antibiotic resistance and its pleiotropic fitness cost

Understanding the conditions that favour the evolution and maintenance of antibiotic resistance is the central goal of epidemiology. A crucial feature explaining the adaptation to harsh, or 'sink', environments is the supply of beneficial mutations via migration from a 'source' population. Given that antibiotic resistance is frequently associated with antagonistic pleiotropic fitness costs, increased migration rate is predicted not only to increase the rate of resistance evolution but also to increase the probability of fixation of resistance mutations with minimal fitness costs. Here we report in vitro experiments using the nosocomial pathogenic bacterium Pseudomonas aeruginosa that support these predictions: increasing rate of migration into environments containing antibiotics increased the rate of resistance evolution and decreased the associated costs of resistance. Consistent with previous theoretical work, we found that resistance evolution arose more rapidly in the presence of a single antibiotic than two. Evolution of resistance was also more rapid when bacteria were subjected to sequential exposure with two antibiotics (cycling therapy) compared with simultaneous exposure (bi-therapy). Furthermore, pleiotropic fitness costs of resistance to two antibiotics were higher than for one antibiotic, and were also higher under bi-therapy than cycling therapy, although the cost of resistance depended on the order of the antibiotics through time. These results may be relevant to the clinical setting where immigration is known to be important between chemotherapeutically treated patients, and demonstrate the importance of ecological and evolutionary dynamics in the control of antibiotic resistance.     

Perceived social isolation,evolutionary fitness and health outcomes: a lifespan approach

Sociality permeates each of the fundamental motives of human existence and plays a critical role in evolutionary fitness across the lifespan. Evidence for this thesis draws from research linking deficits in social relationship—as indexed by perceived social isolation (i.e. loneliness)—with adverse health and fitness consequences at each developmental stage of life. Outcomes include depression, poor sleep quality, impaired executive function, accelerated cognitive decline, unfavourable cardiovascular function, impaired immunity, altered hypothalamic pituitary–adrenocortical activity, a pro-inflammatory gene expression profile and earlier mortality. Gaps in this research are summarized with suggestions for future research. In addition, we argue that a better understanding of naturally occurring variation in loneliness, and its physiological and psychological underpinnings, in non-human species may be a valuable direction to better understand the persistence of a ‘lonely’ phenotype in social species, and its consequences for health and fitness.     

Mapping the environmental fitness landscape of a synthetic gene circuit

Gene expression actualizes the organismal phenotypes encoded within the genome in an environment-dependent manner. Among all encoded phenotypes, cell population growth rate (fitness) is perhaps the most important, since it determines how well-adapted a genotype is in various environments. Traditional biological measurement techniques have revealed the connection between the environment and fitness based on the gene expression mean. Yet, recently it became clear that cells with identical genomes exposed to the same environment can differ dramatically from the population average in their gene expression and division rate (individual fitness). For cell populations with bimodal gene expression, this difference is particularly pronounced, and may involve stochastic transitions between two cellular states that form distinct sub-populations. Currently it remains unclear how a cell population's growth rate and its subpopulation fractions emerge from the molecular-level kinetics of gene networks and the division rates of single cells. To address this question we developed and quantitatively characterized an inducible, bistable synthetic gene circuit controlling the expression of a bifunctional antibiotic resistance gene in Saccharomyces cerevisiae. Following fitness and fluorescence measurements in two distinct environments (inducer alone and antibiotic alone), we applied a computational approach to predict cell population fitness and subpopulation fractions in the combination of these environments based on stochastic cellular movement in gene expression space and fitness space. We found that knowing the fitness and nongenetic (cellular) memory associated with specific gene expression states were necessary for predicting the overall fitness of cell populations in combined environments. We validated these predictions experimentally and identified environmental conditions that defined a "sweet spot" of drug resistance. These findings may provide a roadmap for connecting the molecular-level kinetics of gene networks to cell population fitness in well-defined environments, and may have important implications for phenotypic variability of drug resistance in natural settings.     

Nitrogen mineralization,groundwater dynamics,and forest growth on a Minnesota outwash landscape

We measured aboveground biomass and aboveground net primary productivity (ANPP), groundwater depth and fluctuation, andin situ nitrogen (N) mineralization in 13 upland and 4 wetland forest stands at Cedar Creek Natural History Area (CCNHA). The area, in east central Minnesota (45°25 N, 93°10 W), is on well-sorted glacial outwash of very uniform fine sand. Uplands are interspersed with peadands and the area has shallow groundwater. Stands were aggregated into six ecosystem types based on overstory composition: oak, pine-oak, mesic hardwoods, northern white-cedar, lowland hardwoods, and savanna. Aboveground overstory biomass ranged from 35 to 250 Mg ha^–1; lowest in the savanna and highest in the pine-oak. The ANPP ranged from about 2 to 7.5 Mg ha^–1; also lowest in the savanna but highest in the white-cedar. Over all types, the annual aboveground uptake of N was poorly related to available N measured byin situ mineralization (r ² = 0.01), but the relationship was better (r ² = 0.88) if N availability in the wetland stands was assumed to be a fixed proportion of N in the surface soil (1.5%). Over all types,in situ N mineralization was poorly related to ANPP (r ² = 0.05) and biomass (r ² = 0.38). Both ANPP and overstory biomass were more closely related to groundwater fluctuation (r ² = 0.87 and 0.28, respectively) than to depth (r ² = 0.01 and 0.21, respectively)). The strength of all relationships varied with the inclusion or exclusion of data from the wetland types or the savanna. Total soil N and rates of mineralization were inversely related (r² = 0.42) because of data from wetland stands. Results demonstrate that the positive relationships between aboveground productivity and measuredin situ N mineralization observed in upland forests are not valid for the landscape that includes wetland forests either becausein situ measurements do not indicate N availability in wetlands or because of the presence of other limiting factors. The north temperate landscape includes an abundance of wetland forests with potentially strong linkages to uplands. This study suggests that the commonly-used measure of N availability provides inconsistent information about controls on ecosystems processes in this diverse landscape.Abbreviations ANPP aboveground net primary productivity - CCNHA Cedar Creek National History Area     

Extracting characteristic properties of fitness landscape from in vitro molecular evolution: a case study on infectivity of fd phage to E.coli

We have developed a methodology for extracting characteristic properties of a fitness landscape of interest by analyzing fitness data on an in vitro molecular evolution. The in vitro evolution is required to be conducted as the following "adaptive walk": a single parent sequence generates N mutant sequences as its offsprings, and the fittest individual among the N offsprings will become a new parent in the next generation. N is the library size of mutants to be screened in a single generation. Our theory of the adaptive walk on the "NK landscape" suggests the following: the adaptive walker starting from a random sequence climbs the landscape easily in an early stage, and then reaches a stationary phase in which the mutation-selection-random drift balance sets in. The stationary fitness value is nearly proportional to square root of ln N. Our analysis is performed from the following points: (1) stationary fitness values, (2) time series of fitness in the transitional state, (3) mutant's fitness distribution, and (4) the strength of selection pressure. Applying our methodology, we analyzed experimental data on the in vitro evolution of a random polypeptide (139 amino acids) toward acquiring infectivity (= ability to infect) of fd phage. As a result, we estimated that k is about 27 in this system, indicating that an arbitrary residue in a sequence is affected from other 23% residues. In this article, we demonstrated that the experimental data is consistent with our theoretical equations quantitatively, and that our methodology for extracting characteristic properties of a fitness landscape may be effective.     

Stickbreaking: a novel fitness landscape model that harbors epistasis and is consistent with commonly observed patterns of adaptive evolution

In relating genotypes to fitness, models of adaptation need to both be computationally tractable and qualitatively match observed data. One reason that tractability is not a trivial problem comes from a combinatoric problem whereby no matter in what order a set of mutations occurs, it must yield the same fitness. We refer to this as the bookkeeping problem. Because of their commutative property, the simple additive and multiplicative models naturally solve the bookkeeping problem. However, the fitness trajectories and epistatic patterns they predict are inconsistent with the patterns commonly observed in experimental evolution. This motivates us to propose a new and equally simple model that we call stickbreaking. Under the stickbreaking model, the intrinsic fitness effects of mutations scale by the distance of the current background to a hypothesized boundary. We use simulations and theoretical analyses to explore the basic properties of the stickbreaking model such as fitness trajectories, the distribution of fitness achieved, and epistasis. Stickbreaking is compared to the additive and multiplicative models. We conclude that the stickbreaking model is qualitatively consistent with several commonly observed patterns of adaptive evolution.